The use of positron emission tomography (PET) is growing in the field of medical imaging. In PET imaging, a radiopharmaceutical agent is introduced into the object to be imaged via injection, inhalation, or ingestion. After administration of the radiopharmaceutical, the physical and bio-molecular properties of the agent will cause it to concentrate at specific locations in the human body. The actual spatial distribution of the agent, the intensity of the region of accumulation of the agent, and the kinetics of the process from administration to eventually elimination are all factors that may have clinical significance. During this process, a positron emitter attached to the radiopharmaceutical agent will emit positrons according to the physical properties of the isotope, such as half-life, branching ratio, etc.
The radionuclide emits positrons, and when an emitted positron collides with an electron, an annihilation event occurs, wherein the positron and electron are destroyed. Most of the time an annihilation event produces two 511 keV gamma rays traveling at substantially 180 degrees apart.
By detecting the two gamma rays, and drawing a line between their locations, i.e., the line-of-response (LOR), one can retrieve the likely location of the original disintegration. While this process will only identify a line of possible interaction, by accumulating a large number of those lines, and through a tomographic reconstruction process, the original distribution can be estimated. In addition to the location of the two scintillation events, if accurate timing (within few hundred picoseconds) is available, a time-of-flight (TOF) calculation can add more information regarding the likely position of the event along the line. The collection of a large number of events creates the necessary information for an image of an object to be estimated through tomographic reconstruction. Two detected events occurring at substantially the same time at corresponding detector elements form a line-of-response that can be histogrammed according to their geometric attributes to define projections, or sinograms to be reconstructed.
In PET image reconstruction, interpolation is sometimes used to obtain a uniformly sampled sinogram from a non-uniformly sampled raw LOR sinogram for the purpose of using FBP (filtered back-projection), 3DRP (re-projection), and FORE (Fourier re-binning) reconstruction algorithms. Usually, such multi-dimensional interpolation is performed based on the values on the uniformly sampled rectangular grid points. However, in a PET system having polygon-shaped detector modules arranged in a ring, the raw LOR sinogram sampling points are not on a rectangular grid, but on diamond-patterned sampling grids, as shown in FIG. 1. These diamond-patterned sampling grids can be considered as scattered sampling points.
An interpolation method that has been used in some applications for scattered data is the so-called linear triangular interpolation method (2D), which is known as the linear tetrahedral method in 3D. In the linear triangular interpolation method, 2D triangles are first formed from the scattered sampling points. For example, in Delaunay triangulation, the triangles are formed by connecting all the neighboring points in the Voronoi diagram of a given point set. Once all of the triangles are determined, each sampling point of the uniformly sampled sinogram would fall within (or on the side of) a particular triangle. Further, the value at the new sampling points is then determined by fitting a linear surface on the triangle using barycentric interpolation. A uniformly sampled grid for an interpolated sinogram, which is determined based on the needs of the reconstruction process, is shown in FIG. 2.
However, problems encountered in interpolating a raw LOR sinogram include: (1) difficulty in determining the nearest neighbors of the desired sampling points since they are not on a rectangular grid; and (2) a lack of a systematic method to deal with irregularly distributed sampling points.
Further, linear triangular interpolation is non-trivial to implement. For example, when forming a triangle, none of the raw sampling points can be inside any other triangles. Since irregular, scattered data triangulation is not unique, it is desired to find an optimal triangulation that produces triangles with the largest minimum angle. Further, the triangulation of the sampling data points can be sensitive to the units used for each variable. Even the Delaunay method, which is an optimal triangulation method, can produce different sets of triangulation in some cases.